GOV. Jerry Brown has a golden opportunity to lead by example, to show us he’s serious about putting independent, financially sophisticated people in charge of public pension systems that manage investments worth tens of billions of dollars.

The governor controls the appointments of six of the 12 members of the board of the California State Teachers’ Retirement System, the second-largest public employee pension plan in the nation.

One of those, his finance director, Ana Matosantos, serves by virtue of her job in the administration. Of the other five, two seats are vacant and the others will be at the end of the year.

Whom Brown chooses to appoint will tell us a great deal about whether he plans not only to talk the talk but to walk the walk.

As part of his 12-point pension reform plan released last month, the governor called for increasing pension board independence and expertise “to ensure that retirement funds deliver promised retirement benefits over the long haul without exposing taxpayers to large unfunded liabilities.”

We couldn’t agree more. CalSTRS provides an excellent example of what happens when a pension board’s forecasts are driven by politics rather than sound data, and state lawmakers then fail to provide adequate funding.

The retirement system is funded from three sources. Each year, teachers contribute 8 percent of their salary toward their pensions, while school districts kick in 8.25 percent. The state adds about 2.5 percent, for a total of about 18.75 percent of payroll. However, that barely covers the cost of the benefits those workers earn that year.

It does nothing to address the system’s unfunded liability. Even by its own optimistic calculations, the system has only 71 percent of needed funds, with a $56 billion shortfall. The state would need to kick in an additional $3.7 billion a year for the next 30 years to make the system whole again.

Lawmakers should be blamed for failing to face the problem and start paying down the debt. But one should also blame the retirement system board for using unrealistic projections of benefit costs and investment returns. It is using unrealistic numbers. Consequently, the shortfall is actually significantly worse than $56 billion.

By understating the shortfall, the board makes it easier for lawmakers to postpone action.

If there’s any hope of fixing pension systems across the state, it must begin with an honest discussion of numbers. That requires appointment of people who fully understand, and can critically analyze, the accounting data. That was one of Brown’s messages in his pension plan. Let’s see if he delivers.